a) We find a vertical asymptote of a rational function such that we find a
rational zeros of the denominator i.e the line x = a is a, vertical asymptote
of a rational function, where a is zero of the denominator.

b) Let s(x) be a rational function

\(\displaystyle{s}{\left({x}\right)}={\frac{{{a}_{{{n}}}{x}^{{{n}}}+{a}_{{{n}-{1}}}{x}^{{{n}-{1}}}+\ldots+{a}_{{{1}}}{x}+{a}_{{{0}}}}}{{{b}_{{{m}}}{x}^{{{m}}}+{b}_{{{m}-{1}}}{x}^{{{m}-{1}}}+\ldots+{b}_{{{1}}}{x}+{b}_{{{0}}}}}}\)

then

1. If n < m, then s(x) has horizontal asymptote y = 0,

2. lf n = m, then s has horizontal asymptote \(\displaystyle{y}={\frac{{{a}_{{{n}}}}}{{{b}_{{{m}}}}}}\)

3. If n > m, then s has no horizontal asymptote.

c) We want to find the denominator in the factored form

\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{5}{x}^{{{2}}}+{3}}}{{{x}^{{{2}}}-{4}}}}\)

Sine, we can express the denominator in the factored form

\(\displaystyle{x}^{{{2}}}-{4}={\left({x}-{2}\right)}{\left({x}+{2}\right)}\)

we can see that zeros 2 and -2, therefore from the part a) the lines x=2 and x=-2 are vertical asymptotes.

We can see that n=2 and m=2 i.e n=m. So, from the part b) we obtain that the horizontal asymptote is y=5.

b) Let s(x) be a rational function

\(\displaystyle{s}{\left({x}\right)}={\frac{{{a}_{{{n}}}{x}^{{{n}}}+{a}_{{{n}-{1}}}{x}^{{{n}-{1}}}+\ldots+{a}_{{{1}}}{x}+{a}_{{{0}}}}}{{{b}_{{{m}}}{x}^{{{m}}}+{b}_{{{m}-{1}}}{x}^{{{m}-{1}}}+\ldots+{b}_{{{1}}}{x}+{b}_{{{0}}}}}}\)

then

1. If n < m, then s(x) has horizontal asymptote y = 0,

2. lf n = m, then s has horizontal asymptote \(\displaystyle{y}={\frac{{{a}_{{{n}}}}}{{{b}_{{{m}}}}}}\)

3. If n > m, then s has no horizontal asymptote.

c) We want to find the denominator in the factored form

\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{5}{x}^{{{2}}}+{3}}}{{{x}^{{{2}}}-{4}}}}\)

Sine, we can express the denominator in the factored form

\(\displaystyle{x}^{{{2}}}-{4}={\left({x}-{2}\right)}{\left({x}+{2}\right)}\)

we can see that zeros 2 and -2, therefore from the part a) the lines x=2 and x=-2 are vertical asymptotes.

We can see that n=2 and m=2 i.e n=m. So, from the part b) we obtain that the horizontal asymptote is y=5.